Many dynamical systems combine behaviors ROBUST STABILITY AND CONTROL FOR SYSTEMS THAT COMBINE CONTINUOUS-TIME AND DISCRETE-TIME DYNAMICS

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1 RAFAL GOEBEL, RICARDO G. SANFELICE, and ANDREW R. TEEL ROBUST STABILITY AND CONTROL FOR SYSTEMS THAT COMBINE CONTINUOUS-TIME AND DISCRETE-TIME DYNAMICS Digital Object Identifier.9/MCS Many dynamical systems combine behaviors that are typical of continuous-time dynamical systems with behaviors that are typical of discrete-time dynamical systems. For example, in a switched electrical circuit, voltages and currents that change continuously according to classical electrical network laws also change discontinuously due to switches opening or closing. Some biological systems behave similarly, with continuous change during normal operation and discontinuous change due to an impulsive stimulus. Similarly, velocities in a multibody system change continuously according to Newton s second law but undergo instantaneous changes in velocity and momentum due to collisions. Embedded systems and, more generally, systems involving both digital and analog components form another class of examples. Finally, modern control algorithms often lead to both kinds of behavior, due to either digital components used in implementation or logic and decision making encoded in the control algorithm. These examples fit into the class of hybrid dynamical systems, or simply hybrid systems. This article is a tutorial on modeling the dynamics of hybrid systems, on the elements of stability theory for hybrid systems, and on the basics of hybrid control. The presentation and selection of material is oriented toward the analysis of asymptotic stability in hybrid systems and the design of stabilizing hybrid controllers. Our emphasis on the robustness of asymptotic stability to data perturbation, external disturbances, and measurement error distinguishes the approach taken here from other 8 IEEE CONTROL SYSTEMS MAGAZINE» APRIL X/9/$5. 9IEEE

2 w NESTOR DUB approaches to hybrid systems. While we make some connections to alternative approaches, this article does not aspire to be a survey of the hybrid system literature, which is vast and multifaceted. The interaction of continuous- and discrete-time dynamics in a hybrid system leads to rich dynamical behavior and phenomena not encountered in purely continuous-time systems. Consequently, several challenges are encountered on the path to a stability theory for hybrid systems and to a methodology for robust hybrid control design. The approach outlined in this article addresses these challenges, by using mathematical tools that go beyond classical analysis, and leads to a stability theory that unifies and extends the theories developed for continuous- and discrete-time systems. In particular, we give necessary and sufficient Lyapunov conditions for asymptotic stability in hybrid systems, show uniformity and robustness of asymptotic stability, generalize the invariance principle to the hybrid setting and combine it with Barabasin-Krasovskii techniques, and show the utility of such results for hybrid control design. Despite their necessarily more technical appearance, these results parallel what students of nonlinear systems are familiar with. We now present some background leading up to the model of hybrid systems used in this article. A widely used model of a continuous-time dynamical system is the # first-order differential equation x 5 f x, with x belonging to an n-dimensional Euclidean space Rn. This model can be expanded in two directions that are relevant for hybrid systems. First, we can consider differential equations with # state constraints, that is, x 5 f x and x [ C, where C is a subset of Rn. For example, the set C might indicate that the force of gravity cannot push a ball through the floor. Alternatively, the set C might indicate a set of physically meaningful initial conditions of the system. Second, we can consider the situation where the right-hand side of the differential equation is replaced by a set that may depend on x. For example, when the force applied to a particle varies with time in an unknown way in the interval 3 a, b 4, we can model the derivative of the velocity as belonging to 3 a, b 4. Another reason for considering set-valued righthand sides is to account for the effect of perturbations, such as measurement error in a feedback control system, on a differential equation. Both situations lead to the dif# ferential inclusion x [ F x, where F is a set-valued mapping. APRIL 9 «IEEE CONTROL SYSTEMS MAGAZINE 9

3 The examples of hybrid control systems provided in this article only scratch the surface of what is possible using hybrid feedback control. Combining the two generalizations leads to constrained differential inclusions x # [ Fx, x [ C. A typical model of a discrete-time dynamical system is the first-order equation x 5 gx, with x [ R n. The notation x indicates that the next value of the state is given as a function of the current state x through the value gx. As for differential equations, it is a natural extension to consider constrained difference equations and difference inclusions, which leads to the model x [ Gx, x [ D, where G is a set-valued mapping and D is a subset of R n. Since a model of a hybrid dynamical system requires a description of the continuous-time dynamics, the discretetime dynamics, and the regions on which these dynamics apply, we include both a constrained differential inclusion and a constrained difference inclusion in a general model of a hybrid system in the form x # [ F(x), x [ C, () x [ G(x), x [ D. () The model (), () captures a wide variety of dynamic phenomena including systems with logic-based state components, which take values in a discrete set, as well as timers, counters, and other components. Examples in this article demonstrate how to cast hybrid automata and switched systems, as well as sampled-data and networked control systems, into the form (), (). We refer to a hybrid system in the form (), () as H. We call C the flow set, F the flow map, D the jump set, and G the jump map. For many systems, the generality provided by the inclusions in (), () is not needed. Thus, the reader may ν p p (a) ν (c) FIGURE Collision between two particles. (a) Two particles are initialized to positions p and p and with velocities v and v. (b) An impact between the particles occurs at the position p *. (c) The direction of the motion of each particle is reversed after the impact. p (b) replace the set-valued mappings and the corresponding inclusions in (), () with equations and proceed confidently. It is often the geometry of sets C and D that produces the rich dynamical phenomena in a hybrid system rather than the multivaluedness of the mappings F and G. However, this article does justify, beyond the sake of generality, the use of differential inclusions and difference inclusions. We provide examples of hybrid models in the following section. Subsequently, we make precise the meaning of a solution to a hybrid dynamical system and describe basic mathematical properties of the space of solutions. Afterward, we present results on asymptotic stability in hybrid systems, with an emphasis on robustness. Initially, we focus on Lyapunov functions as the primary stability analysis tool and show how Lyapunov functions are used in hybrid control design. Finally, we present tools for stability analysis based on limited events in hybrid systems and show how these tools are related to hybrid feedback control algorithms. The main developments of the article are complemented by several supporting discussions. Hybrid Automata and Switching Systems relate systems in the form (), () to hybrid automata and switching systems, respectively. Related Mathematical Frameworks presents other mathematical descriptions of systems where features of both continuous- and discretetime dynamical systems are present. Existence, Uniqueness, and other Well-Posedness Issues discusses basic properties of solutions to hybrid systems in the form (), (). Set Convergence and Robustness and Generalized Solutions introduce mathematical tools from beyond classical analysis and motivate the assumptions placed on the data of a hybrid system H, as given by C, F, D, G. Motivating Stability of Sets and Why Pre - Asymptotic Stability? explain distinct features of the asymptotic stability concept used in the article. Converse Lyapunov Theorems and Invariance state and discuss main tools used in the stability analysis. Zeno Solutions describes a phenomenon unique to hybrid dynamical systems. Simulation in Matlab/ Simulink presents an approach to simulation of hybrid systems. The notation used throughout this article is defined in List of Symbols. 3 IEEE CONTROL SYSTEMS MAGAZINE» APRIL 9

4 HYBRID PHENOMENA AND MODELING Colliding Masses Many engineering systems experience impacts [8], [56]. Walking and jumping robots, juggling systems, billiards, and a bouncing ball are examples. Continuous-time equations of motion describe the behavior of these systems between impacts, whereas discrete dynamics approximate what happens during impacts. Consider two particles that move toward each other, collide, and then move away from each other, as shown in Figure. Before and after the collision, the position and velocity of each particle are governed by Newton s second law. At impact, the velocity evolution is modeled by an instantaneous change in the velocities but no change in the positions of the particles. The combined continuous and discrete behavior of the particles can be modeled as a hybrid system with a differential equation governing the continuous dynamics, constraints describing where the continuous dynamics apply, a difference equation governing the discrete dynamics, and constraints describing where the discrete dynamics apply. The state is the vector x 5 p, v 5 p, p, v, v, where p and p denote the particles positions and v and v denote the particles velocities. The state vector changes continuously if, as shown in Figure, the first particle s position is at or to the left of the second particle s position. This condition is described by the flow set C J 5p, v : p # p 6. Assuming no friction, the flow map obtained applying Newton s second law and using () is Fp, v 5 v,, v,. An impact occurs when the positions of the particles are identical and their velocities satisfy v $ v. These conditions define the jump set D J 5p, v : p 5 p, v $ v 6. Letting v and v indicate the velocities after an impact, we have the conservation of momentum equation m v m v 5 m v m v (3) and the energy dissipation equation v v 5rv v, (4) where m and m are the masses of the particles and the constant r [, is a restitution coefficient. Solving (3) and (4) for v and v and using () yields the jump map Gp, p, v, v 5 p, p, v m lv v, v m lv v, where l5 r / m m. Impulsive Behavior in Biological Systems Synchronization in groups of biological oscillators occurs in swarms of fireflies [], groups of crickets [88], ensembles of List of Symbols x # The derivative, with respect to time, of the x R R n R $ state of a hybrid system The state of a hybrid system after a jump The set of real numbers The n -dimensional Euclidean space The set of nonnegative real numbers, R $ 5 3, ` ) Z The set of all integers N The set of nonnegative integers, N 5 5,, c6 N $k 5k, k, c6 for a given k [ N [ The empty set S The closure of the set S con S The convex hull of the set S con S The closure of the convex hull of a set S S \ S The set of points in S that are not in S S 3 S The set of ordered pairs (s, s ) with s [ S, s [ S x^ The transpose of the vector x (x, y) Equivalent notation for the vector 3x^y^4 T x B The Euclidean norm of a vector x [ R n The closed unit ball, of appropriate dimension, in the Euclidean norm x S inf y [g x y for a set S ( R n and a point x [ R n S n The set 5x [ R n : x 5 6 f : R m S R n F : R m S R n R(# ) A function from R m to R n A set-valued mapping from R m to R n The rotation matrix Rf 5 c cosf sinf sinf cosf d F(S) h x[s F(x) for the set-valued mapping F : R m S R n and a set S ( R m T S (h) The tangent cone to the set S ( R n at h [ S. T S (h) is the set of all vectors w [ R n for K` which there exist h i [ S, t i., for all i 5,, c such that h i S h, t i R, and (h i h) /t i S w as i S ` The class of functions from R $ to R $ that are continuous, zero at zero, strictly increasing, and unbounded L V (m) The m-level set of the function V : dom V S R, which is the set of points 5x [ d om V : V(x) 5m6 neuronal oscillators [3], and groups of heart muscle cells [6]. Detailed treatments include [6] and [78]. The discussion below is related to [] and [55], where models of a collection of nonlinear clocks with impulsive coupling are studied. A model of two linear clocks with impulsive coupling is used in [6] to analyze the synchronization of heart muscle cells. APRIL 9 «IEEE CONTROL SYSTEMS MAGAZINE 3

5 Hybrid Automata Some models of hybrid systems explicitly partition the state of a system into a continuous state j and a discrete state q, the latter describing the mode of the system. For example, the values of q may represent modes such as working and idle. In a temperature control system, q may stand for on or off, while j may represent the temperature. By its nature, the discrete state can change only during a jump, while the continuous state often changes only during fl ows but sometimes may jump as well. These systems are called differential automata [8], hybrid automata [S], [5], or simply hybrid systems [S], [9]. All of these systems can be cast as a hybrid system of the form (), (). The data of a hybrid automaton are usually given by» a set of modes Q, which in most situations can be identified with a subset of the integers» a domain map Domain: Q S R n, which gives, for each q [ Q, the set Domain(q) in which the continuous state j evolves» a flow map f : Q 3 R n S R n, which describes, through a differential equation, the continuous evolution of the continuous state variable j» a set of edges Edges ( Q 3 Q, which identifies the pairs (q, qr) such that a transition from the mode q to the mode qr is possible» a guard map Guard : Edges S R n, which identifies, for each edge (q, qr) [ Edges, the set Guardq, qr to which the continuous state j must belong so that a transition from q to qr can occur» a reset map Reset : Edges 3 R n S R n, which describes, for each edge (q, qr) [ Edges, the value to which the continuous state j [ R n is set during a transition from mode q to mode qr. When the continuous state variable j remains constant at a jump from q to qr, the map Resetq, qr, # can be taken to be the identity. Figure S depicts part of a state diagram for a hybrid automaton. The continuous dynamics of two modes are shown, together with the guard conditions and reset rules that govern transitions between these modes. We now show how a hybrid automaton can be modeled as a hybrid system in the form (), (). First, we reformulate a hybrid automaton as a hybrid system with explicitly shown modes. For each q [ Q, we take C q 5 Domain(q), D q 5 d Guard(q, qr), (q, qr)[edges F q (j) 5 f(q, j), for all j [ C q, G q (j) 5 d 5qr:j[Guard(q, qr)6 (Reset(q, qr, j), qr), for all j [ D q. When j is an element of two different guard sets Guard(q, qr) and Guard(q, qs ), G q (j) is a set consisting of at least two points. Hence, G q can be set valued. In fact, G q is not necessarily a function even when every Reset(q, qr, # ) is the identity map. With C q, F q, D q, and G q defi ned above, we consider the hybrid system with state (j, q) [ R n 3 R and representation j # 5 F q (j), q [ Q, j [ C q, (j, q ) [ G q (j), q [ Q, j [ D q. Example S: Reformulation of a Hybrid Automaton Consider the hybrid automaton shown in figures S and S3, with the set of modes Q 5 5, 6 ; the domain map given by Domain() 5 R # 3 R, Domain() R; the fl ow map, for all j [ R, given by f(, j) 5 (, ), f(, j) 5 (,); the set of edges given by Edges 5 5 (, ), (, ), (, ) 6 ; the guard map given by ξ Guard(q, q ) ξ + Reset(q, q, ξ) q.. q ξ = f (q, ξ ) ξ = f (q, ξ ) ξ Domain(q) ξ Domain(q ) Guard(, ) 5 R $ 3 R #, Guard(, ) 5 R $, Guard(, ) R # ; and the reset map, for all j[ R, given by ξ + Reset(q, q, ξ) ξ Guard(q, q) Reset(,, j) 5 (5, ), Reset(,, j) 5j, Reset(,, j) 5 j. FIGURE S Two modes, q and qr, of a hybrid automaton. In mode q, the state j evolves according to the differential equation j # 5 f(q, j) in the set Domainq. A transition from mode q to mode qr can occur when, in mode q, j is in the set Guardq, qr. During the transition, j changes to a value j in Resetq, qr, j. Transitions from mode q to other modes, not shown in the figure, are governed by similar rules. The sets Guard(, ) and Guard(, ) overlap, indicating that, in mode, a reset of the state j to (5, ) or a switch of the mode to is possible from points 3 IEEE CONTROL SYSTEMS MAGAZINE» APRIL 9

6 where j $ and j 5. Formulating this hybrid automaton as a hybrid system with explicitly shown modes leads to ξ R R ξ + = ( 5, ) ξ R ξ + = ξ D 5 Guard, c Guard, 5 R $ 3 R and the set- valued q = jump map G given by. q =. ξ = (, ) ξ = (, ) (5,, ), if j $, j,, ξ R R ξ {} R G (j) 5 (5,, ) h (j, j, ), if j $, j 5, (j, j, ), if j $, j.. ξ + = ξ ξ {} R FIGURE S Two modes of the hybrid automaton in Example S. In mode, the state j evolves according to the differential equation j # 5 (, ) in the set Domain() 5 R # 3 R. To formulate a hybrid automaton in the form (), (), we define x 5 (j, q) [ R n and A transition from mode to mode occurs when j is in the guard set Guard(, ) 5 R $ but not in the guard set C 5 dq[q (C q 3 5q6), F(x) 5F q (j) 3 56 for all x [ C, Guard(, ) 5 R $ 3 R #. During the transition, j does not change its value. Also in mode, a jump in j to the value (5, ) D 5 dq[q (D q 3 5q6), G(x) 5G q (j) for all x [ D. occurs when j is in the guard set Guard(, ) 5R $ 3 R # but not in the guard set Guard(, ) 5R $. When j belongs to Guard(, ) and Guard(, ), either the transition to mode When the domains and guards are closed sets, the fl ow and or the jump of j can occur. In mode, the state j evolves according to the differential equation j # 5 (, ) in the set jump sets C and D are also closed. Similarly, when the fl ow and Domain() R. A transition from mode to mode can reset maps are continuous, the fl ow map F and the jump map G occur when j is in the guard set Guard(, ) R #. During satisfy the Basic Assumptions. the transition, j changes to the value j. Example S Revisited: Solutions to a Hybrid Automaton Consider the hybrid system modeling the hybrid automaton of Example S. The initial condition j5(3, 3), in mode q 5, of the hybrid automaton corresponds to the initial condition (3, 3, ) for the hybrid system. The maximal solution to the hybrid system starting from (3, 3, ), denoted x a, has domain dom x a 5, c, and is given by x a (, ) 5 (3, 3, ), x a (, ) 5 (3, 3, ). The solution jumps once, the jump takes x a outside of both the jump set and the fl ow set, and thus cannot be extended. The hybrid system has multiple solutions from the initial point (,, ). One maximal solution starting from (,, ), denoted x b, is complete and never flows. This solution has dom x b N and is given by x b (, j) 5.5.5() j. That is, the solution x b switches back and forth between mode and mode infinitely many times. Another solution starting from (,, ), denoted x c, has dom x c 5, c 3, 54, c 35, 4, c, 3 and is given by x c (t, j) 5 µ (,, ), if t 5, j 5, (5 t, t, ), if t [ 3, 54, j 5, (, 5 (t 5), ), if t [ 35, 4, j 5, (,, ), if t 5, j 5 3. In the language of hybrid automata, this solution undergoes a reset of the state without a switch of the mode, fl ows for fi ve units of time until it hits a guard, switches the mode without resetting the state, fl ows for another fi ve units of time until it hits another guard, and switches the mode without resetting the state. This solution is not maximal since it can be extended 5 q = q = x Domain() x Guard(, ) Domain() Guard(, ) x x Guard(, ) FIGURE S3 Data for the hybrid automaton in Example S. Solid arrows indicate the direction of flow in Domain() and Domain(). Dashed arrows indicate jumps from Guard(, ) and Guard(, ). in several ways. One way is by concatenating x c with x b, that is, by setting x c (t, j 3) 5x b (t, j) for (t, j) [ dom x b. In other words, x c can be extended by back and forth switches between modes. The solution x c can be also extended to be periodic. We can consider x c (t, j) 5 (5 (t ), t, ) for t [ 3, 54, j 5 4, x c (t, j) 5 (, 5 (t 5), ) for t [ 35, 4, s 5 5, x c (, 6) 5 (,, ), and repeat. REFERENCES [S] M. S. Branicky, V. S. Borkar, and S. K. Mitter, A unified framework for hybrid control: Model and optimal control theory, IEEE Trans. Automat. Contr., vol. 43, no., pp. 3 45, 998. [S] T. A. Henzinger, The theory of hybrid automata, in Proc. of the th Annu. Symp. Logic in Computer Science, IEEE CS Press, 996, pp APRIL 9 «IEEE CONTROL SYSTEMS MAGAZINE 33

7 Switching Systems A switching system is a differential equation whose right-hand side is chosen from a family of functions based on a switching signal [49], [S4]. For each switching signal, the switching system is a time-varying differential equation. As in [S3], we study the properties of a switching system not under a particular switching signal but rather under various classes of switching signals. In the framework of hybrid systems, information about the class of switching signals often can be embedded into the system data by using timers and reset rules, which can be viewed as an autonomous, that is, time-invariant hybrid subsystem. Results for switching systems, including converse Lyapunov theorems and invariance principles [5], can be then deduced from results obtained for hybrid systems. A switched system can be written as j # 5 f q (j), where, for each q [ Q 5 5,, c, q max 6, f q : R n S R n (S) is a continuous function. A complete solution to the system (S) consists of a locally absolutely continuous function j : R $ S R n and a function q : R $ S Q that is piecewise constant, has a fi nite number of discontinuities in each compact time interval, and satisfi es j # (t) 5 f q(t) (j(t)) for almost all t [ R $. In what follows, given a complete solution (j, q) to (S), let I be the number of discontinuities of q, with the possibility of I 5`, and I let t 5 and 5t i 6 i5 be the increasing sequence of times at which q is discontinuous. For simplicity, we discuss complete solutions only. A solution (j, q) to (S) is a dwell-time solution with dwell time t D. if t i t i $t D for all i 5,, c, I. That is, switches are separated by at least an amount of time t D. Each dwell-time solution can be generated as part of a solution to the hybrid system with state x 5 (j, q, t) [ R n given by j 3 j 4 3 t t t 3 t (a) t t t 3 t (b) FIGURE S4 Hybrid time domain for a solution under dwell-time switching and average dwell-time switching. (a) Hybrid time domain for a dwell-time solution with dwell-time constant t D larger or equal than min5t t, t 3 t, c6. (b) Hybrid time domain for an average dwell-time solution for parameters (d, N) satisfying the average dwell-time condition in (S). For example, parameters d, N 5 4/min 5t, t t 6, and (d, N) 5 (4/t, 4) satisfy (S). The domain repeats periodically, as denoted by the blue dot. Consider a group of fireflies, each of which has an internal clock state. Suppose each firefly s clock state increases monotonically until it reaches a positive threshold, assumed to be the same for each firefly. When a firefly s clock reaches its threshold, the clock resets to zero and the firefly flashes, which causes the other fireflies clocks to jump closer to their thresholds. In this way, the flash of one firefly affects the internal clocks of the other fireflies. Figure depicts the evolution of the internal clocks of two fireflies coupled through flashes. The time units are normalized so that each firefly s internal clock state takes values in the interval 3, 4, and thus the threshold for flashing is one for each firefly. A hybrid model for a system of n fireflies, each with the same clock characteristics, has the state x 5 x, c, x n, flow map Fx 5 f x, c, f x n, where f : 3, 4 S R. is continuous, and flow set C 5 3, 4 n, where 3, 4 n indicates the set of points x in R n for which each component x i belongs to the interval 3, 4. The function f governs the rate at which each clock state evolves in the interval 3, 4. Since jumps in the state of the system are allowed when any one of the fireflies clocks reaches its threshold, the jump set is D 5 5x [ C : max i x i 5 6. One way to model the impulsive changes in the clock states is through a rule that instantaneously advances a clock state by a factor e, where e., when this action does not take the clock state past its threshold. Otherwise, the clock state is reset to zero, just as if it had reached its threshold. The corresponding jump map G does not satisfy the regularity condition (A3) of the Basic Assumptions imposed in the section Basic Mathematical Properties. The algorithm for defining generalized solutions in Robustness and 34 IEEE CONTROL SYSTEMS MAGAZINE» APRIL 9

8 j # 5 f q j q # 5 5: Fx, x [ C J R n 3 Q 3 3, 4, t # [ 3, /t D 4 j 5 j q [ Q 5: Gx, x [ D J R n 3 Q t 5 Note that it takes at least an amount of time t D for the timer state t to increase from zero to one with velocity t # [ 3, /t D 4. Therefore, t ensures that jumps of this hybrid system occur with at least t D amount of time in between them. In fact, there is a one-to-one correspondence between dwell-time solutions to (S) and solutions to the hybrid system initialized with t5 for which t # 5 when t [ 3, ) and t # 5 when t5. A solution (j, q) to (S) is an average dwell-time solution if the number of switches in a compact interval is bounded by a number that is proportional to the length of the interval, with proportionality constant d $, plus a positive constant N [34]. In the framework of hybrid systems, each average dwell-time solution has a hybrid time domain E such that, for each pair (s, i) and (t, i) belonging to E and satisfying with s # t and i # j, j i # (t s)dn. (S) Dwell-time solutions are a special case, corresponding to d5/t D and N 5. Every hybrid time domain that satisfi es (S) can be generated by the hybrid subsystem with compact fl ow and jump sets given by t # [ 3, d4, t [ 3, N4, t 5t, t [ 3, N4. (S3) (S4) The time domain for each solution of this hybrid system satisfies the constraint (S). Furthermore, for every hybrid time domain E satisfying (S) there exists a solution of (S3), (S4), starting at t5n, and defined on E [S5], [4]. In turn, switching systems under an average dwell-time constraint with parameters (d, N) are captured by the hybrid system with state x 5 (j, q, t) [ R n given by j # 5 f q j q # 5 5: Fx, x [ C J R n 3 Q 3 3, N4, t # [ 3, d4 j 5 j q [ Q 5: Gx, x [ D J R n 3 Q 3 3, N4. t 5 t Figure S4 depicts a hybrid time domain for a dwell-time solution and an average dwell-time solution to a switching system. More elaborate families of solutions to switching systems can be modeled by means of hybrid systems. We briefly mention one such family. A solution (x, q) to (S) is a persistent dwell-time solution with persistent dwell time t D. and period of persistence T. if there are infinitely many intervals of length at least t D on which no switches occur, and such intervals are separated by at most an amount of time T [S3]. A hybrid system that models such solutions involves two timers. One timer ensures that periods with no switching last at least an amount of time t D ; the other timer ensures that periods of arbitrary switching do not last more than an amount of time T. The hybrid system also involves a differential inclusion j # [ F(j), where Fj J con h q[q f q j, to describe solutions to the switching system during periods of arbitrary switching. REFERENCES [S3] J. P. Hespanha, Uniform stability of switched linear systems: Extensions of LaSalle s invariance principle, IEEE Trans. Automat. Contr., vol. 49, pp , 4. [S4] D. Liberzon and A. S. Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. Mag., vol. 9, pp. 59 7, 999. [S5] S. Mitra and D. Liberzon, Stability of hybrid automata with average dwell time: An invariant approach, in Proc. 43rd IEEE Conf. Decision and Control, Bahamas, Dec. 4, pp Generalized Solutions motivates the modified jump map Gx 5 gx, c, gx n, where gx i 5 ex i, when ex i,,, when ex i., 5, 6, when ex i 5, which does satisfies the regularity condition (A3). This jump map advances the clock state x i by the factor e when this action keeps x i below the threshold, and it resets x i to zero when multiplying x i by e produces a value greater than the threshold value. Either resetting the clock state to zero or advancing the clock state by e can occur when ex i 5. A group of fireflies can exhibit almost global synchronization, meaning that, from almost every initial condition, the state vector tends to the set where all of the clock states are equal [55]. Synchronization analysis for the case n 5 is given in Example 5. Explicit Zero-Crossing Detection Zero-crossing detection (ZCD) algorithms for sinusoidal signals are crucial for estimating phase and frequency as well as power factor in electric circuits. ZCD algorithms employ a discrete state, which remembers the most recent sign of the signal and is updated when the signal crosses zero, as indicated in Figure 3. We cast a simple ZCD algorithm for a sinusoidal signal in terms of a hybrid system. Let the sinusoid be generated as the output of the linear system j # 5 vj, j # 5 vj, y 5 j, where v., and let q denote a discrete state taking values in Q J 5, 6 corresponding to the sign of j. The state of the hybrid system is x 5 j, q, while the flow map is Fx 5 vj, vj,. APRIL 9 «IEEE CONTROL SYSTEMS MAGAZINE 35

9 x ξ t x t q t (a) t ξ FIGURE Trajectories of the internal clocks of two fireflies with impulsive coupling. When either clock state x or x reaches the unit threshold, both states experience a jump. When a state reaches the threshold, it is reset to zero. At the same time, the other state is increased by a factor e if this increase does not push the state past the threshold; otherwise, this state is also reset to zero. ξ As j changes sign, a zero-crossing event occurs. We model the detection of this event as a toggling of the state q through the rule q 5q. In a more elaborate model, either a counter that keeps track of the number of zero-crossing events can be incremented, or a timer state that monitors the amount of time between zero-crossing events can be reset. The state j does not change during jumps. The jump map is thus Gx 5 j, q. When q and j have the same sign, that is, qj $, flows are allowed. This behavior corresponds to the flow set C 5 h q[q C q 3 5q6, where C q J 5j [ R : qj $ 6. In other words, the flow set C is the union of two sets. One set corresponds to points where q 5 and j $, while the other set corresponds to points where q 5 and j #. When j 5 and the sign of q is opposite to the sign of the derivative of j, that is, qj #, the value of q is toggled. This behavior corresponds to the jump set D 5 h q[q D q 3 5q6, where D q J 5j [ R : j 5, qj # 6. Thus, the jump set D is the union of two sets. One set corresponds to points where q 5, j 5, and j #, while the other set corresponds to points where q 5, j 5, and j $. Figure 4 shows the flow and jump sets of the hybrid system. The figure also depicts the sinusoidal signal j and (b) FIGURE 3 Detection of zero crossings of a sinusoidal signal. The sinusoidal signal j is the output of the linear system j # 5 vj, j # 5 vj, where v.. (a) The discrete state q is toggled at every zero crossing of the sinusoidal signal j. (b) The signal evolves in the j5(j, j ) plane. the discrete state q obtained for initial conditions with j starting at one, j starting at zero, and q starting at one. Sample-and-Hold Control Systems In a typical sample-and-hold control scenario, a continuous-time plant is controlled by a digital controller. The controller samples the plant s state, computes a control signal, and sets the plant s control input to the computed value. The controller s output remains constant between updates. Sample-and-hold devices perform analog-to-digital and digital-to-analog conversions. As noted in [59], the closed-loop system resulting from this control scheme can be modeled as a hybrid system. Sampling, computation, and control updates in 36 IEEE CONTROL SYSTEMS MAGAZINE» APRIL 9

10 sample-and-hold control are associated with jumps that occur when one or more timers reach thresholds defining the update rates. When these operations are performed synchronously, a single timer state and threshold can be used to trigger their execution. In this case, a sample-andhold implementation of a control law samples the state of the plant and updates its input when a timer reaches the threshold T., which defines the sampling period. During this update, the timer is reset to zero. For the static, state-feedback law u 5kj for the plant j # 5 fj, u, a hybrid model uses a memory state z to store the samples of u, as well as a timer state t to determine when each sample is stored. The state of the resulting closed-loop system, which is depicted in Figure 5, is taken to be x 5 j, z, t. During flow, which occurs until t reaches the threshold T, the state of the plant evolves according to j # 5 fj, z, the value of z is kept constant, and t grows at the constant rate of one. In other words, z # 5 and t # 5. This behavior corresponds to the flow set C 5 R n 3 R m 3 3, T4, while the flow map is given by Fx 5 fj, z,, for all x [ C. When the timer reaches the threshold T, the timer state t is reset to zero, the memory state z is updated to kj, but the plant state j does not change. This behavior corresponds to the jump set D J R n 3 R m 3 5T6 and the jump map Gx J j, kj, for all x [ D. q = q = ξ ξ D ξ ξ (, ) C D C FIGURE 4 Flow and jump sets for each q [ Q and trajectory to the hybrid system in Explicit Zero-Crossing Detection. The trajectory starts from the initial condition at (t, j) 5 (, ) given by j (, ) 5, j (, ) 5, q(, ) 5. The jumps occur on the j axis and toggle q. Flows are permitted in the left-half plane for q 5 and in the right-half plane for q 5. A hybrid controller is defined by a flow set C c ( R nm, flow map f c : C c S R n, jump set D c ( R nm, and a possibly set-valued jump map G c : R nm S R m, together with a feedback law k c : C c S R r that specifies the control signal u. Figure 6 illustrates this setup. During continuous-time evolution, which can occur when the composite closed-loop state x 5 x p, x c belongs to the set C c, the controller state satisfies x # c 5 f c x and the control signal is generated as u 5k c x. At jumps, which are allowed when the closed-loop state belongs to D c, the state of the controller is reset using the rule x c [ G c x. The closed-loop system is a hybrid system with state x 5 x p, x c, flow set C 5 C c, jump set D 5 D c, flow map ξ Hybrid Controllers for Nonlinear Systems Hybrid dynamical systems can model a variety of closedloop feedback control systems. In some hybrid control applications the plant itself is hybrid. Examples include juggling [7], [73] and robot walking control [63]. In other applications, the plant is a continuous-time system that is controlled by an algorithm employing discrete-valued states. This type of control appears in a broad class of industrial applications, where programmable logic controllers and microcontrollers are employed for automation. In these applications, discrete states, as well as other variables in software, are used to implement control logic that incorporates decision-making capabilities into the control system. Consider a plant described by the differential equation x # p 5 f p x p, u, (5) where x p [ R n, u [ R r, and f p is continuous. A hybrid controller for this plant has state x c [ R m, which can contain logic states, timers, counters, observer states, and other continuous-valued and discrete-valued states. and jump map Fx 5 c f p x p, k c x d for all x [ C, (6) f c x ZOH u = z T D/A Nonlinear System Algorithm A/D FIGURE 5 Digital control of a continuous-time nonlinear system with sample-and-hold devices performing the analog-to-digital (A/D) and digital-to-analog (D/A) conversions. Samples of the state j of the plant and updates of the control law k(j) computed by the algorithm are taken after each amount of time T. The controller state z stores the values of k(j). ξ T APRIL 9 «IEEE CONTROL SYSTEMS MAGAZINE 37

11 The interaction of continuous- and discrete-time dynamics in a hybrid system leads to rich dynamical behavior and phenomena not encountered in purely continuous-time systems. x p Gx 5 c d for all x [ D. (7) G c x One way hybrid controllers arise is through supervisory control. A supervisor oversees a collection of controllers and makes decisions about which controller to insert into the closed-loop system based on the state of the plant and the controllers. The supervisor associates to each controller a region of operation and a region where switching to other controllers is possible. These regions are subsets of the state space. In the region where changes between controllers are allowed, the supervisor specifies the controllers to which authority can be switched. In supervisory control, it is possible for the individual controllers to be hybrid controllers. Through this degree of flexibility, it is possible to generate hybrid control algorithms through a hierarchy of supervisors. The following example features a supervisor for two state-feedback control laws. Example : Dual-Mode Control for Disk Drives Control of read/write heads in hard disk drives requires precise positioning on and rapid transitioning between tracks on a disk drive. To meet these dual objectives, some u = K c (x p, x c ) Nonlinear System Plant State: x p (Continuous State) Controller State: x c (Such as Timers and Discrete States) FIGURE 6 Closed-loop system consisting of a continuous-time nonlinear system and a hybrid controller. The nonlinear system has state x p, which is continuous, and input u. The hybrid controller has state x c, which has continuous state variables, such as timer states, and discrete state variables, such as logic modes. The control input u 5k c (x p, x c ) to the nonlinear system is a function of the plant state x p and the controller state x c. H c x p commercial hard disk drives use mode-switching control [7], [8], [87], which combines a track-seeking controller and a track-following controller. The track-seeking controller rapidly steers the magnetic head to a neighborhood of the desired track, while the track-following controller regulates position and velocity, precisely and robustly, to enable read/write operations. Mode- switching control uses the track-seeking controller to steer the magnetic head s state to a point where the track-following controller is applicable, and then switches the control input to the track-following controller. The control strategy results in a hybrid closed-loop system. Let p [ R be the position and v [ R the velocity of the magnetic head in the disk drive. The dynamics can be approximated by the double integrator system p # 5 v, v # 5 u [7], [87]. The hybrid controller for the magnetic head supervises both the track-seeking control law u 5k p, v, p* and the track-following control law u 5k p, v, p*, where p* is the desired position. We assume that the track-seeking control law globally asymptotically stabilizes the point p*,, while the track-following control law locally asymptotically stabilizes the point p*,. Let C be a compact neighborhood of p*, that is contained in the basin of attraction for p*, when using the track-following control law, and let D be a compact neighborhood of p*, such that solutions using the track-following control law that start in D do not reach the boundary of C. Also define C 5 R \D and D 5 R \C. Figure 7 illustrates these sets. Let the controller state q [ Q J 5, 6 denote the operating mode. The track-seeking mode corresponds to q 5, while the track-following mode corresponds to q 5. The mode-switching strategy uses the track-seeking controller when p, v [ C and the track-following controller when p, v [ C. Figure 7 indicates the intersection of C and C, where either controller can be used. To prevent chattering between the two controllers in the intersection of C and C, the supervisor allows mode switching when p, v [ D q. In other words, a switch from the track-seeking mode to the track-following mode can occur when p, v [ D, while a switch from the track-following mode to the track-seeking mode can occur when p, v [ D. The hybrid controller executing this logic has the flow set C c 5 h q[q C q 3 5q6, flow map f c p, v, q 5, jump set 38 IEEE CONTROL SYSTEMS MAGAZINE» APRIL 9

12 D c J h q[q D q 3 5q6, and jump map G c p, v, q 5 3 q, which toggles q in the set Q 5 5, 6. The idea behind this control construction applies to arbitrary nonlinear control systems and state-feedback laws [66]. CONCEPT OF A SOLUTION D C B Generalized Time Domains Solutions to continuous-time dynamical systems are para meterized by t [ R $, whereas solutions to discretetime dynamical systems are parameterized by j [ N. Parameterization by t [ R $ is possible for a continuous-time system even when solutions experience jumps, as long as at most one jump occurs at each time t. For example, parameterization by t [ R $ is used for switched systems [49] as well as for impulsive dynamical systems [43], [3]. However, parameterization by t [ R $ of discontinuous solutions to a dynamical system may be an obstacle for establishing sequential compactness of the space of solutions. For example, sequential compactness may require us to admit a solution with two jumps at the same time instant to represent the limit of a sequence of solutions for which the time between two consecutive jumps shrinks to zero. By considering a generalized time domain, we can overcome such obstacles and, furthermore, treat continuous- and discrete-time systems in a unified framework. A subset E of R $ 3 N is a hybrid time domain [3], [6] if it is the union of infinitely many intervals of the form 3t j, t j 4 3 5j6, where 5 t # t # t # c, or of finitely many such intervals, with the last one possibly of the form 3t j, t j 4 3 5j6, 3t j, t j 3 5j6, or 3t j, ` 3 5j6. An example of a hybrid time domain is shown in Figure 8. A hybrid time domain is called a hybrid time set in [7] and is equivalent to a generalized time domain [5] defined as a sequence of intervals, some of which may consist of one point. The idea of a hybrid time domain is present in the concept of a solution given in [4], which explicitly includes a nondecreasing sequence of jump times in the solution description. More general time domains are sometimes considered. For details, see [54], [8], or the discussion of time scales in Related Mathematical Frameworks. Some time domains make it possible to continue solutions past infinitely many jumps. For an initial exposition of hybrid system and for the analysis of many hybrid control algorithms, domains with this feature are not necessary. Solutions A solution to a hybrid system is a function, defined on a hybrid time domain, that satisfies the dynamics and constraints given by the data of the hybrid system. The data in (), () has four components, which are the flow set, the flow map, the jump set, and the jump map. For j 3 t = t (p, ) FIGURE 7 Sets of the hybrid controller for dual-mode control of disk drives. The flow and jump sets for the track-seeking mode q 5 and the track-following mode q 5 are constructed from the sets C, D and C, D, respectively. The set B is the basin of attraction for (p *, ) when the track-following controller is applied. In addition, this set contains the compact set D, from which solutions with the track-following controller do not reach the boundary of C, a compact subset of B. This property is illustrated by the solid black solution starting from D. The dashed black solution is the result of applying the track-seeking controller, which steers solutions to the set D in finite time. FIGURE 8 A hybrid time domain. The hybrid time domain, which is denoted by E, is given by the union of 3, t , 3t, t , 3t, t , and 3t 3, t 4 ) a hybrid system (), () on R n, the flow set C is a subset of R n, the flow map is a set-valued mapping F : R n S S R n, the jump set D is a subset of R n, and the jump map is a set-valued mapping G : R n S S R n. A set-valued mapping on R n associates, to each x [ R n, a set in R n. A function is a set-valued mapping whose values can be viewed as sets that consist of one point. A hybrid arc is a function x : dom x S R n, where dom x is a hybrid time domain and, for each t 3 C D E t 4 t APRIL 9 «IEEE CONTROL SYSTEMS MAGAZINE 39

13 Related Mathematical Frameworks Interest in hybrid systems grew rapidly in the 99s with computer scientists and control systems researchers coming together to organize several international workshops. See [S8] and similar subsequent collections. Additional books dedicated to hybrid systems include [86] and [S]. The legacy of the cooperative initiative with computer science is the successful conference Hybrid Systems: Computation and Control (HSCC), now a part of the larger cyber-physical systems week, which includes real-time and embedded systems and information processing in sensor networks. At the same time, many mathematical frameworks related to hybrid systems have also appeared in the literature. Some of those frameworks are discussed below. Additional ideas appear in the concept of a discontinuous dynamical system, described in [S] and [S]. IMPULSIVE DIFFERENTIAL EQUATIONS Impulsive differential equations consist of the classical differential equation x # (t) 5 f(x(t)), which applies at all times except the impulse times, and the equation Dx(t i ) 5 I i (x(t i )), which describes the impulsive behavior at impulse times. The impulse times are often fi xed a priori for each particular solution and form an increasing sequence t, t, c. In other words, a solution with the state x(t i ) before the ith jump has the value x(t i ) I i (x(t i )) after the jump. Solutions to impulsive differential equations are piecewise differentiable or piecewise absolutely continuous functions parameterized by time t. These functions cannot model multiple jumps at a single time instant. An impulsive differential equation can be recast as a hybrid system in the case where the impulse times are determined by the condition x(t) [ D for some set D. This situation requires some conditions on D and I i to ensure that x(t i ) I i (x(t i )) o D. For simplicity, consider the case where I i is the same map I for each i. Then the corresponding hybrid system has the flow map f, the jump map x A x I(x), the jump set D, and the flow set C given by the complement of D. Natural generalizations of impulsive differential equations include impulsive differential inclusions, where either f or I may be replaced by a set-valued mapping. For details, see [S6], [43], [S5], and [3]. MEASURE-DRIVEN DIFFERENTIAL EQUATIONS The classical differential equation x # (t) 5 f(x(t)) can be rewritten as dx(t) 5 f(x(t))dt. Measure-driven differential equations are formulated as dx(t) 5 f (x(t))dt f (x(t))m(dt), where f, f are functions and m is a nonnegative scalar or vector-valued Borel measure. Solutions to measure-driven differential equations are given by functions of bounded variation parameterized by t and are not necessarily differentiable, absolutely continuous, or even continuous. The discontinuous behavior is due to the presence of atoms in the measure m. In control situations, the driving measure m, in particular, the atoms of m, can approximate time-dependent controls that take large values on short intervals. C D The hybrid arc x is a solution to the hybrid system H 5 C, F, D, G if x, [ C h D and the following conditions are satisfied. x Flow Condition For each j [ N such that I j has nonempty interior, x # t, j [ Fxt, j for almost all t [ I j, xt, j [ C for all t [ 3min I j, sup I j. FIGURE 9 Evolution of a solution to a hybrid system. Flows and jumps of the solution x are allowed only on the flow set C and from the jump set D, respectively. The solid blue curves indicate flow. The dashed red arcs indicate jumps. The solid curves must belong to the flow set C. The dashed arcs must originate from the jump set D. fixed j, t A xt, j is a locally absolutely continuous function on the interval I j 5 5t : t, j [ dom x6. Jump Condition For each t, j [ dom x such that t, j [ dom x, x(t, j ) [ G(x(t, j)), x(t, j) [ D. If the flow set C is closed and I j has nonempty interior, then the requirement xt, j [ C for all t [ 3min I j, sup I j in the flow condition is equivalent to xt, j [ C for 4 IEEE CONTROL SYSTEMS MAGAZINE» APRIL 9

14 Natural generalizations, needed to analyze mechanical systems with friction or impacts [S3], include measuredriven differential inclusions, where f, f are replaced by setvalued mappings. Formulating a robust notion of a solution to measure-driven differential inclusions is technically challenging [S9], [S4]. DYNAMICAL SYSTEMS ON TIME SCALES A framework for unifying the classical theories of differential and difference equations is that of dynamical systems on time scales [S7]. Given a time scale T, which is a nonempty closed subset of R, a generalized derivative of a function f : T S R relative to T can be defi ned. This generalized derivative reduces to the standard derivative when T 5 R, and to the difference f(n ) f(n) when evaluated at n and for T 5 N. As special cases, classical differential and difference equations can be written as systems on time scales. Systems on time scales can also be used to model populations that experience a repeated pattern consisting of continuous evolution followed by a dormancy [S7, Ex..39]. Consider a time scale T that is unbounded to the right, and, for t [ T, define s(t) 5 inf 5s [ T : s. t6. The generalized derivative of f : T S R at t [ T is the number f D (t), if it exists, such that, for each e. and each s [ T sufficiently close to t, 3f(s(t))f(s) 4 f D (t) 3s(t) s4 #es(t) s. The function f is differentiable if f D exists at every t [ T. A dynamical system on the time scale T has the form x D (t) 5 f(x(t)) for every t [ T. One advantage of the framework of dynamical systems on time scales is the generality of the concept of a time scale. A drawback, especially for control engineering purposes, is that a time scale is chosen a priori, and all solutions to a system are defined on the same time scale. REFERENCES [S6] D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: Stability, Theory, and Applications. Chichester, U.K.: Ellis Horwood, 989. [S7] M. Bohner and A. Peterson, Dynamic Equations on Time Scales. Cambridge, MA: Birkhaüser,. [S8] R. L. Grossman, A. Nerode, A. P. Ravn, and H. Rischel, Eds. Hybrid Systems. New York: Springer-Verlag, 993. [S9] G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differ. Integr. Equ., vol. 4, pp , 99. [S] A. S. Matveev and A. V. Savkin, Qualitative Theory of Hybrid Dynamical Systems. Cambridge, MA: Birkhaüser,. [S] A. N. Michel, L. Wang, and B. Hu, Qualitative Theory of Dynamical Systems: The Role of Stability Preserving Mappings. New York: Marcel Dekker,. [S] A. N. Michel, L. Hou, and D. Liu, Stability of Dynamical Systems: Continuous, Discontinuous, and Discrete Systems. Cambridge, MA: Birkhaüser, 8. [S3] J.-J. Moreau, Unilateral contact and dry friction in finite freedom dynamics, in Non-smooth Mechanics and Applications, New York: Springer-Verlag, 988, pp. 8. [S4] G. N. Silva and R. B. Vinter, Measure driven differential i n c l u s i o n s, J. M a t h. A n a l. A p p l i c a t., v o l., n o. 3, p p , 996. [ S5 ] T. Yang, Impulsive Control Theory. Berlin : Springer- Verlag,. all t [ I j and is also equivalent to xt, j [ C for almost all t [ I j. The solution x to a hybrid system is nontrivial if dom x contains at least one point different from, ; maximal if it cannot be extended, that is, the hybrid system has no solution xr whose domain dom xr contains dom x as a proper subset and such that xr agrees with x on dom x; and complete if dom x is unbounded. Every complete solution is maximal. Figure 9 shows a solution to a hybrid system flowing, as solutions to continuous-time systems do, at points in the flow set C, and jumping, as solutions to discretetime systems do, from points in the jump set D. At points where D overlaps with the interior of C, solutions can either flow or jump. Thus, the jump set D enables rather than forces jumps. To force jumps from D, the flow set C can be replaced by either the set C \ D or the set C \ D. The parameterization of a solution x by t, j [ dom x means that xt, j represents the state of the hybrid system after t time units and j jumps. Figure shows a j 3 x (, ) x (t, j) t = t t 3 t 4 t dom x FIGURE A solution to a hybrid system. The solution, which is denoted by x, has initial condition x(, ), is given by a hybrid arc, and has hybrid time domain dom x. The hybrid time domain E in Figure 8 is equal to dom x. APRIL 9 «IEEE CONTROL SYSTEMS MAGAZINE 4